3.523 \(\int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=137 \[ \frac {a^3 \sin ^7(c+d x)}{7 d}+\frac {a^3 \sin ^6(c+d x)}{2 d}+\frac {a^3 \sin ^5(c+d x)}{5 d}-\frac {5 a^3 \sin ^4(c+d x)}{4 d}-\frac {5 a^3 \sin ^3(c+d x)}{3 d}+\frac {a^3 \sin ^2(c+d x)}{2 d}+\frac {3 a^3 \sin (c+d x)}{d}+\frac {a^3 \log (\sin (c+d x))}{d} \]

[Out]

a^3*ln(sin(d*x+c))/d+3*a^3*sin(d*x+c)/d+1/2*a^3*sin(d*x+c)^2/d-5/3*a^3*sin(d*x+c)^3/d-5/4*a^3*sin(d*x+c)^4/d+1
/5*a^3*sin(d*x+c)^5/d+1/2*a^3*sin(d*x+c)^6/d+1/7*a^3*sin(d*x+c)^7/d

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Rubi [A]  time = 0.11, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac {a^3 \sin ^7(c+d x)}{7 d}+\frac {a^3 \sin ^6(c+d x)}{2 d}+\frac {a^3 \sin ^5(c+d x)}{5 d}-\frac {5 a^3 \sin ^4(c+d x)}{4 d}-\frac {5 a^3 \sin ^3(c+d x)}{3 d}+\frac {a^3 \sin ^2(c+d x)}{2 d}+\frac {3 a^3 \sin (c+d x)}{d}+\frac {a^3 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*Log[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Sin[c + d*x]^2)/(2*d) - (5*a^3*Sin[c + d*x]^3)/(3*d)
- (5*a^3*Sin[c + d*x]^4)/(4*d) + (a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(2*d) + (a^3*Sin[c + d*x]^
7)/(7*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps

\begin {align*} \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a (a-x)^2 (a+x)^5}{x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^5}{x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (3 a^6+\frac {a^7}{x}+a^5 x-5 a^4 x^2-5 a^3 x^3+a^2 x^4+3 a x^5+x^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {a^3 \log (\sin (c+d x))}{d}+\frac {3 a^3 \sin (c+d x)}{d}+\frac {a^3 \sin ^2(c+d x)}{2 d}-\frac {5 a^3 \sin ^3(c+d x)}{3 d}-\frac {5 a^3 \sin ^4(c+d x)}{4 d}+\frac {a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^6(c+d x)}{2 d}+\frac {a^3 \sin ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 88, normalized size = 0.64 \[ \frac {a^3 \left (60 \sin ^7(c+d x)+210 \sin ^6(c+d x)+84 \sin ^5(c+d x)-525 \sin ^4(c+d x)-700 \sin ^3(c+d x)+210 \sin ^2(c+d x)+1260 \sin (c+d x)+420 \log (\sin (c+d x))\right )}{420 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(420*Log[Sin[c + d*x]] + 1260*Sin[c + d*x] + 210*Sin[c + d*x]^2 - 700*Sin[c + d*x]^3 - 525*Sin[c + d*x]^4
+ 84*Sin[c + d*x]^5 + 210*Sin[c + d*x]^6 + 60*Sin[c + d*x]^7))/(420*d)

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fricas [A]  time = 0.64, size = 112, normalized size = 0.82 \[ -\frac {210 \, a^{3} \cos \left (d x + c\right )^{6} - 105 \, a^{3} \cos \left (d x + c\right )^{4} - 210 \, a^{3} \cos \left (d x + c\right )^{2} - 420 \, a^{3} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (15 \, a^{3} \cos \left (d x + c\right )^{6} - 66 \, a^{3} \cos \left (d x + c\right )^{4} - 88 \, a^{3} \cos \left (d x + c\right )^{2} - 176 \, a^{3}\right )} \sin \left (d x + c\right )}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/420*(210*a^3*cos(d*x + c)^6 - 105*a^3*cos(d*x + c)^4 - 210*a^3*cos(d*x + c)^2 - 420*a^3*log(1/2*sin(d*x + c
)) + 4*(15*a^3*cos(d*x + c)^6 - 66*a^3*cos(d*x + c)^4 - 88*a^3*cos(d*x + c)^2 - 176*a^3)*sin(d*x + c))/d

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giac [A]  time = 0.28, size = 108, normalized size = 0.79 \[ \frac {60 \, a^{3} \sin \left (d x + c\right )^{7} + 210 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} - 525 \, a^{3} \sin \left (d x + c\right )^{4} - 700 \, a^{3} \sin \left (d x + c\right )^{3} + 210 \, a^{3} \sin \left (d x + c\right )^{2} + 420 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 1260 \, a^{3} \sin \left (d x + c\right )}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/420*(60*a^3*sin(d*x + c)^7 + 210*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c)^5 - 525*a^3*sin(d*x + c)^4 - 700*a
^3*sin(d*x + c)^3 + 210*a^3*sin(d*x + c)^2 + 420*a^3*log(abs(sin(d*x + c))) + 1260*a^3*sin(d*x + c))/d

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maple [A]  time = 0.47, size = 144, normalized size = 1.05 \[ -\frac {a^{3} \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7 d}+\frac {176 a^{3} \sin \left (d x +c \right )}{105 d}+\frac {22 a^{3} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35 d}+\frac {88 a^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{105 d}-\frac {a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{2 d}+\frac {\left (\cos ^{4}\left (d x +c \right )\right ) a^{3}}{4 d}+\frac {a^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)*(a+a*sin(d*x+c))^3,x)

[Out]

-1/7/d*a^3*cos(d*x+c)^6*sin(d*x+c)+176/105*a^3*sin(d*x+c)/d+22/35/d*a^3*cos(d*x+c)^4*sin(d*x+c)+88/105/d*a^3*c
os(d*x+c)^2*sin(d*x+c)-1/2/d*a^3*cos(d*x+c)^6+1/4/d*cos(d*x+c)^4*a^3+1/2/d*a^3*cos(d*x+c)^2+a^3*ln(sin(d*x+c))
/d

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maxima [A]  time = 0.51, size = 107, normalized size = 0.78 \[ \frac {60 \, a^{3} \sin \left (d x + c\right )^{7} + 210 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} - 525 \, a^{3} \sin \left (d x + c\right )^{4} - 700 \, a^{3} \sin \left (d x + c\right )^{3} + 210 \, a^{3} \sin \left (d x + c\right )^{2} + 420 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 1260 \, a^{3} \sin \left (d x + c\right )}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/420*(60*a^3*sin(d*x + c)^7 + 210*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c)^5 - 525*a^3*sin(d*x + c)^4 - 700*a
^3*sin(d*x + c)^3 + 210*a^3*sin(d*x + c)^2 + 420*a^3*log(sin(d*x + c)) + 1260*a^3*sin(d*x + c))/d

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mupad [B]  time = 8.99, size = 178, normalized size = 1.30 \[ \frac {176\,a^3\,\sin \left (c+d\,x\right )}{105\,d}-\frac {a^3\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a^3\,{\cos \left (c+d\,x\right )}^2}{2\,d}+\frac {a^3\,{\cos \left (c+d\,x\right )}^4}{4\,d}-\frac {a^3\,{\cos \left (c+d\,x\right )}^6}{2\,d}+\frac {88\,a^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{105\,d}+\frac {22\,a^3\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{35\,d}-\frac {a^3\,{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )}{7\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^5*(a + a*sin(c + d*x))^3)/sin(c + d*x),x)

[Out]

(176*a^3*sin(c + d*x))/(105*d) - (a^3*log(1/cos(c/2 + (d*x)/2)^2))/d + (a^3*log(sin(c/2 + (d*x)/2)/cos(c/2 + (
d*x)/2)))/d + (a^3*cos(c + d*x)^2)/(2*d) + (a^3*cos(c + d*x)^4)/(4*d) - (a^3*cos(c + d*x)^6)/(2*d) + (88*a^3*c
os(c + d*x)^2*sin(c + d*x))/(105*d) + (22*a^3*cos(c + d*x)^4*sin(c + d*x))/(35*d) - (a^3*cos(c + d*x)^6*sin(c
+ d*x))/(7*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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